Robust quantification of the exposure to operational risk: Bringing economic sense to economic capital Alberto Suárez, Santiago Carrillo EPS, Universidad Autónoma de Madrid (Spain) RiskLab Madrid alberto.suarez@uam.es 1
Please, ask questions! 2
Basel II http://www.bis.org/publ/bcbs128.htm Robust estimation of OR measures 3
The three pillars approach � First pillar : Minimum capital requirements (quantification of risk) • Specifies the guiding principles for the estimation of regulatory/economic capital. • Operational risk is included as a new type of risk. � Second pillar: Supervisory review process. � Third pillar: Market discipline (+ public disclosure) Robust estimation of OR measures 4
Operational Risk: Definition [source: Basel II] 644. Operational risk is defined as the risk of loss resulting from inadequate or failed internal processes , people and systems or from external events . This definition includes legal risk, but excludes strategic and reputational risk. Robust estimation of OR measures 5
Operational Risk: Measurement Robust estimation of OR measures 6
Measurement approaches � Basic Indicator Approach (BIA) � α = 0.15 � = α × K EI, where EI=gross income (mean of the last 3 years) BIA � � Standardised Approach (TSA) � β are defined by the regulator 8 � i � = β × EI , where EI are the gross income for line i. K TSA i i � = i 1 i � Advanced Measurement Approaches (AMA) � Scorecard approach. � Loss Distribution approach. Robust estimation of OR measures 7
AMA Soundness Standard (Basel II) 667. Given the continuing evolution of analytical approaches for operational risk, the Committee is not specifying the approach or distributional assumptions used to generate the operational risk measure for regulatory capital purposes. However, a bank must be able to demonstrate that its approach captures potentially severe ‘tail’ loss events. Whatever approach is used, a bank must demonstrate that its operational risk measure meets a soundness standard comparable to that of the internal ratings-based approach for credit risk, (i.e. comparable to a one year holding period and a 99.9th percentile confidence interval). Robust estimation of OR measures 8
Business lines & risk types Risk type Business ��������� ��������� ����������� ���������� Execution, ����������� ����������� ��������� ��������� line �������������� ��������� Delivery & ������ ������ ��������� ����������� ���������������� ������ � Process ��������� �������� Management Corporate Finance Trading & Sales Retail Banking Commercial Banking Payment and Settlement Agency Services and Custody Asset Management Retail Brokerage Robust estimation of OR measures 9
Loss distribution approach � Model the distribution of the aggregate losses for a given business line & risk type [ i , j ] N � t [ i , j ] [ i , j ] = Loss X ; t nt = n 1 [ i , j ] is the number of losses in year N t t for business line and risk type . i j � Calculate Capital at Risk (99.9% percentile) of the aggregate loss distribution per 8 7 �� [ i , j ] = CaR CaR business line & risk type = = and add them. i 1 j 1 Robust estimation of OR measures 10
Actuarial models: Frequency + Severity � Hypothesis � Severities of losses are independent � Severities and frequencies are independent � Model separately � Frequency { N t } E.g. Poisson, negative binomial, Cox process,… � Severity { X nt } E. g. Lognormal, Gaussian inverse, Gamma, Weibull, … � Obtain the distribution of aggregate losses by combining these distributions. [ Panjer , FFT , MC sim.] Robust estimation of OR measures 11
Risk analysis � Calculate aggregate yearly loss distribution from the frequency and severity distributions. � Compute risk measures • Expected loss • Capital at Risk ( CaR ) e.g. 99.9% percentile of the aggregate loss distribution. • Conditional CaR ( Expected shortfall) Expected loss, given that the loss is above CaR Robust estimation of OR measures 12 12
Aggregation of frequency and severity dists. Robust estimation of OR measures 13
Expected & unexpected loss Robust estimation of OR measures 14
Computational issues in risk analysis Algorithms to compute risk measures � Deterministic algorithms Discretized approximation to aggregate loss distribution. • Panjer • Fast Fourier Transform (FFT) � Monte Carlo algorithms Empirical compound distribution obtained by simulation. Computationally costly. • Use variance reduction techniques • Hardware solutions: Grid computing, GPU’s, … Robust estimation of OR measures 15 15
Modeling the frequency of events � Use only internal data � Time unit for fit : 1 day, 1week, 1 month, 1 year (too few data!) � Model distributions: � Poisson � Negative binomial � Cox process � Empirical The differences between the risk measures obtained with different models are generally small. � Correct the distribution parameters to take into account the collection threshold . Robust estimation of OR measures 16 16
Model distributions for frequencies � Poisson model One-parameter model average frequency: λ mean = variance � Negative binomial Two parameters mean < variance Robust estimation of OR measures 17 17
Modeling the severity of events � Use internal + external + scenarios � Take into account the collection threshold in the fit (truncated data) � Model distributions: � Lognormal � Piecewise models: • Model for the body (e.g. empirical, lognormal) • Model for the tail • Generalized Pareto • g-and-h distribution The differences among the risk measures obtained with different models are generally large . Robust estimation of OR measures 18 18
Modeling the severity of events Focus on the tail Robust estimation of OR measures 19 19
Separate models for the body and tail Robust estimation of OR measures 20 20
A cautionary tale Tails are notoriously difficult to model Robust estimation of OR measures 21
The lognormal distribution � � 1 1 ( ) 2 � � µ σ = − − µ LNpdf ( x ; , ) exp log x � � 2 σ 2 2 πσ 2 x > x 0 µ + σ X ~ exp( Z ); Z ~ N ( 0 , 1 ) � exp( µ ) � scale � σ � tails Robust estimation of OR measures 22
The g-and-h distribution � � gZ − 1 1 e � � 2 = + X a b exp h Z � � g 2 Z ~ N ( 0 , 1 ) � g � skewness � h � kurtosis Advantages � Flexible, realistic fits (Dutta & Perry, 2007 ) Disadvantages � Slow convergence to asymptotic regime (EVT) (Degen et al., 2007) � Unstable estimates of parameters Robust estimation of OR measures 23
Extreme Value Theory and operational risk � Asymptotic regime : CaR is dominated by single extreme events from the tail of the severity distribution. � Asymptotically, the tail of a distribution is has Generalized Pareto form. � These extreme events should be • Independent. • Identically distributed. • Constant probability occurrence per unit time. � � Poisson distribution. � � � Model : Poisson + Pareto tail . Robust estimation of OR measures 24
The Generalized Pareto distribution � Probability density function β ξ GPpdf ( x ; u , , ) 1 − − 1 � � ξ 1 ξ ( ) � � = + − 1 x u � � + � � β β β > 0 ≥ ξ ≥ x u (if 0) Robust estimation of OR measures 25
The parameter ξ � If ξ ≥ 0.5 the variance diverges. � If ξ ≥ 1 the mean diverges. � The expected loss is not defined. � Empirical estimates of the unexpected loss (the difference between a high percentile of the aggregate loss distribution and the expected loss) can be negative ! � In Pareto fits to empirical operational loss data, values of ξ ξ close to 1 and even larger can be found. ξ ξ Robust estimation of OR measures 26
Pareto fit: Estimates of ξ (N = 10 5 ) � Theoretical value for Pareto data ξ ξ = 0.7 ξ ξ � Theoretical value for lognormal data ξ ξ ξ ξ = 0 Robust estimation of OR measures 27
Pareto fit: Estimates of ξ (N = 10 3 ) � Theoretical value for Pareto data ξ ξ = 0.7 ξ ξ � Asymptotic value for lognormal data ξ ξ ξ ξ = 0 Robust estimation of OR measures 28 28
Sensitivity to single events (N=10 3 , M=100) CaR ( × 10 -3 ) � � u Theoretical 1930 2300 0.7 3604 Maximum 1900 2352 0.55 1144 excluded [1876, 1914] [2086, 2726] [0.45, 0.70] [661, 3167] Maximum 1928 2303 0.66 2492 included [1913, 1.934] [2022, 2619] [0.55, 0.81] [1261, 7695] 32 -65 0.1 1335 variation [19, 50] [-107, -39] [0.08, 0.13] [538, 4537] 1.67 -2.78 17.92 104.01 % variation [0.97, 2.69] [-4.48, -1.72] [13.03, 26.96] [70.36, 145.72] Robust estimation of OR measures 29
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